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The speed of sound greatly exceeds typical flow velocities in many stellarand planetary interiors. To follow the slow evolution of subsonic motions,various sound-proof models attempt to remove fast acoustic waves whilstretaining stratified convection and buoyancy dynamics. In astrophysics,anelastic models typically receive the most attention in the class ofsound-filtered stratified models. Generally, anelastic models remain valid innearly adiabatically stratified regions like stellar convection zones, but maybreak down in strongly sub-adiabatic, stably stratified layers common instellar radiative zones. However, studying stellar rotation, circulation, anddynamos requires understanding the complex coupling between convection andradiative zones, and this requires robust equations valid in both regimes. Herewe extend the analysis of equation sets begun in Brown Vasil & Zweibel 2012,which studied anelastic models, to two types of pseudo-incompressible models.This class of models has received attention in atmospheric applications, andmore recently in studies of white-dwarf supernovae progenitors. We demonstratethat one model conserves energy but the other does not. We use Lagrangianvariational methods to extend the energy conserving model to a general equationof state, and dub the resulting equation set the GeneralizedPseudo-Incompressible (GPI) model. We show that the GPI equations suitablycapture low frequency phenomena in both convection and radiative zones in starsand other stratified systems, and we provide recommendations for convertinglow-Mach number codes to this equation set.

Energy conservation and gravity waves in sound-proof treatments of stellar interiors: Part II Lagrangian constrained analysis